Adaptive Control with Sparse Identification of Nonlinear Dynamics

1. Problem Statement

The paper addresses the challenge of controlling uncertain nonlinear control-affine systems where the system dynamics are not fully known but can be represented as a linear combination of a known, over-complete library of candidate basis functions.

• System Model: $\dot{x} = Y(x)\theta + g(x)u$, where $x$ is the state, $u$ is the control input, $g(x)$ is a known control effectiveness matrix, and $Y(x)$ is a known library of nonlinear basis functions. The parameter vector $\theta$ is unknown and assumed to be sparse (i.e., only a few elements are non-zero, representing the true governing physical laws).
• Challenges:
  1. Parameter Uncertainty: Traditional adaptive control requires Persistent Excitation (PE), a strong condition often difficult to satisfy in practice.
  2. Model Structure: Existing data-driven methods (like SINDy) often operate offline or lack closed-loop stability guarantees when integrated into real-time control.
  3. Sparsity vs. Stability: Integrating sparsity-promoting techniques (like $\ell_1$ regularization) into online adaptive control introduces non-smoothness (due to the absolute value function), complicating stability analysis and the design of projection operators to keep estimates bounded.

2. Methodology

The authors propose a Sparsity-Promoting Integral Concurrent Learning (SP-ICL) framework. This approach combines adaptive control, concurrent learning (using historical data), and sparse identification.

A. Control Architecture

• Tracking Controller: A standard certainty-equivalence feedback controller is used:
  $$u = g^+(x)(\dot{x}_d - Y(x)\hat{\theta} - Ke)$$
  where $x_d$ is the desired trajectory, $e$ is the tracking error, and $K$ is a positive definite gain.
• Error Dynamics: The tracking error dynamics are derived as $\dot{e} = -Ke + Y(x)\tilde{\theta}$, where $\tilde{\theta} = \theta - \hat{\theta}$ is the parameter estimation error.

B. Parameter Estimator (SP-ICL)

The core innovation is the update law for $\hat{\theta}$, which minimizes a regularized cost function:
$$J(\hat{\theta}) = \frac{1}{2}\hat{\theta}^\top Y \hat{\theta} - \hat{\theta}^\top U + \lambda \|\hat{\theta}\|_1$$

• Integral Concurrent Learning (ICL): Instead of relying solely on instantaneous data, the algorithm maintains a history stack of filtered regressor data ($Y_f$) and state data ($U_f$). This allows the system to learn from past data points, relaxing the need for strict PE conditions to a weaker Finite Excitation (FE) condition.
• Sparsity Promotion: An $\ell_1$-regularization term ($\lambda \|\hat{\theta}\|_1$) is added to the cost function to drive irrelevant parameters to zero.
• Non-Smooth Update Law: Since the $\ell_1$ norm is non-differentiable at zero, the update law is formulated as a differential inclusion using the Clarke sub-differential (generalized gradient):
  $$\dot{\hat{\theta}} \in \text{proj}_\Theta(\dots) - \lambda \gamma \Gamma \text{SGN}(\hat{\theta})$$
  where $\text{SGN}$ is the set-valued signum function.
• Data Selection: A specific algorithm (Algorithm 1) manages the history stack to ensure the memory regressor matrix remains full rank (satisfying the FE condition) by replacing old data with new data that maximizes the minimum eigenvalue.

C. Stability Analysis

• Non-Smooth Lyapunov Analysis: Because the update law involves a set-valued map (due to the signum function), standard Lyapunov analysis is insufficient. The authors employ non-smooth Lyapunov analysis (using Filippov solutions and differential inclusions).
• Boundedness: They prove that the parameter estimates remain within a compact set using a projection operator adapted for set-valued maps.
• Ultimate Boundedness: Theorem 1 establishes that under the FE condition and appropriate gain selection, the closed-loop system trajectories are ultimately bounded. The tracking error and parameter estimation error converge to a small neighborhood of the origin.

3. Key Contributions

1. Integration of Sparsity and Control: The paper successfully integrates $\ell_1$-regularization (common in machine learning/SINDy) into a Lyapunov-based adaptive control framework with rigorous stability guarantees.
2. Relaxed Excitation Conditions: By utilizing ICL with a history stack, the method achieves parameter convergence under Finite Excitation (FE) rather than the stricter Persistent Excitation (PE) condition.
3. Handling Non-Smoothness: The authors develop a novel approach to handle the non-smooth nature of the $\ell_1$ penalty within the adaptive loop. They prove that projecting only the continuous part of the update law is sufficient to maintain forward invariance of the parameter estimates, avoiding the complexity of projecting the entire set-valued map.
4. Online Sparse Identification: Unlike offline SINDy, this method identifies the sparse structure of the dynamics in real-time while simultaneously ensuring trajectory tracking.

4. Simulation Results

The authors validated the method on a nonlinear control-affine system with a known sparse ground truth ($\theta$) and a library of 20 candidate basis functions.

• Effect of $\lambda$ (Sparsity Parameter):
  • Low $\lambda$: The model recovers all true terms but includes many false positives (non-sparse).
  • Optimal $\lambda$ ($10^{-2}$): The method correctly identifies the sparse structure with high precision (F1 score of 0.89). It recovers the true dynamics while eliminating spurious correlations.
  • High $\lambda$: Excessive sparsity leads to the removal of true terms (false negatives), degrading model fidelity and increasing tracking error.
• Chattering: The discontinuous signum term induces chattering in the parameter estimates and tracking error. This effect increases with larger $\lambda$.
• Performance Trade-off: There is a clear trade-off between sparsity (model simplicity) and tracking accuracy. Moderate sparsity improves parameter convergence by removing noise, but excessive sparsity harms performance.

5. Significance and Conclusion

This work bridges the gap between data-driven sparse identification (like SINDy) and robust adaptive control.

• Theoretical Impact: It provides a rigorous mathematical framework for using non-smooth regularization in online control, proving stability where previous methods might have failed or lacked guarantees.
• Practical Impact: The SP-ICL framework allows controllers to learn the "true" physical laws of a system (sparse dynamics) in real-time, even with limited excitation. This is crucial for applications like autonomous driving or robotics where friction and environmental conditions change, and accurate, interpretable models are needed for safety and performance.
• Future Work: The authors suggest mitigating chattering by replacing the discontinuous signum function with continuous approximations (e.g., saturation or hyperbolic tangent) and exploring adaptive $\lambda$ tuning to balance sparsity and accuracy dynamically.